Subgroups — #26216661

Order 12032.

1350 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15689145	C₂	1	6	yes	maximal
2	C₄³×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20243540	C₂	1	1	yes	maximal, characteristic, complemented
4	#20243539	C₂	1	6	yes	maximal, complemented
5	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10625324	C₂²	1	4	yes	—
7	C₂×C₄²×D₄₇	C₄	1	3	yes	—
8	C₂×C₄²×D₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄²×D₄₇	C₂²	1	3	yes	—
10	#15689145	—	2	24	no	—
11	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	6	yes	—
12	C₂×Dic₄₇⋊C₈	C₂²	1	12	yes	—
13	C₄²×Dic₄₇	C₄	1	3	yes	—
14	C₄²×Dic₄₇	C₄	1	1	yes	characteristic
15	C₄²×Dic₄₇	C₂²	1	3	yes	—
16	C₂×C₄⋊C₈×C₄₇	C₂²	1	6	yes	—
17	C₄³×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×D₄₇	C₂×C₄	1	3	yes	—
19	C₂²×C₄×D₄₇	C₂×C₄	1	3	yes	—
20	C₂²×C₄×D₄₇	C₂³	1	1	yes	characteristic
21	C₂×#10625324	—	2	24	no	—
22	C₂²×C₄₇⋊C₈	C₂³	1	4	yes	—
23	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
24	C₂×C₄×Dic₄₇	C₂×C₄	1	6	yes	—
25	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
26	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
27	C₂×C₄×Dic₄₇	C₂×C₄	1	3	yes	—
28	C₂×C₄×Dic₄₇	C₂³	1	3	yes	—
29	C₄²×D₄₇	—	2	4	no	—
30	C₄²×D₄₇	—	2	12	no	—
31	C₄²×D₄₇	D₄	1	12	yes	—
32	C₄²×D₄₇	Q₈	1	12	yes	—
33	C₄⋊(C₄₇⋊C₈)	—	2	12	no	—
34	Dic₄₇⋊C₈	—	2	24	no	—
35	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
36	C₂×C₄²×C₄₇	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
39	C₄⋊C₈×C₄₇	—	2	12	no	—
40	C₂³×D₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
42	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
43	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
44	C₂²×Dic₄₇	C₂²×C₄	1	3	yes	—
45	C₂²×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₄₇	C₂²×C₄	1	3	yes	—
47	C₂×C₄×D₄₇	—	2	6	no	—
48	C₂×C₄×D₄₇	—	2	6	no	—
49	C₂×C₄×D₄₇	—	2	12	no	—
50	C₂×C₄×D₄₇	C₄⋊C₄	1	12	yes	—
51	C₂×C₄×D₄₇	C₄⋊C₄	1	12	yes	—
52	C₂×C₄×D₄₇	C₂×D₄	1	6	yes	—
53	C₂×C₄×D₄₇	C₂×Q₈	1	6	yes	—
54	C₂×C₈×C₄₇	—	2	12	no	—
55	C₄²×C₄₇	—	2	6	no	—
56	C₄²×C₄₇	M₁₆	1	4	yes	complemented
57	C₄²×C₄₇	C₂×D₄	1	6	yes	—
58	C₄²×C₄₇	C₂×Q₈	1	6	yes	—
59	C₄×Dic₄₇	—	2	6	no	—
60	C₄×Dic₄₇	—	2	12	no	—
61	C₄×Dic₄₇	C₄⋊C₄	1	12	yes	—
62	C₄×Dic₄₇	C₄⋊C₄	1	12	yes	—
63	C₄×Dic₄₇	M₁₆	1	12	yes	complemented
64	C₄×Dic₄₇	C₂×D₄	1	6	yes	—
65	C₄×Dic₄₇	C₂×Q₈	1	6	yes	—
66	C₂×C₄₇⋊C₈	—	2	12	no	—
67	#10625324	—	4	16	no	—
68	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
69	C₂²×D₄₇	—	2	3	no	—
70	C₂²×D₄₇	—	2	1	no	—
71	C₂²×D₄₇	C₂×C₄⋊C₄	1	6	yes	—
72	C₂×C₄×C₄₇	—	2	6	no	—
73	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
74	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
75	C₂×C₄×C₄₇	C₂×M₁₆	1	6	yes	—
76	C₂×C₄×C₄₇	C₂×M₁₆	1	6	yes	—
77	C₂×C₄×C₄₇	C₂²×D₄	1	3	yes	—
78	C₂×C₄×C₄₇	C₂²×Q₈	1	3	yes	—
79	C₂×Dic₄₇	—	2	3	no	—
80	C₂×Dic₄₇	—	2	3	no	—
81	C₂×Dic₄₇	C₂×C₄⋊C₄	1	6	yes	—
82	C₂×Dic₄₇	C₂×C₄⋊C₄	1	6	yes	—
83	C₂×Dic₄₇	C₂×C₄⋊C₄	1	6	yes	—
84	C₂×Dic₄₇	C₂×M₁₆	1	12	yes	—
85	C₄×D₄₇	—	2	24	no	—
86	C₄×D₄₇	—	2	24	no	—
87	C₄×D₄₇	#127	1	4	yes	—
88	C₄×D₄₇	C₄⋊Q₈	1	12	yes	—
89	C₈×C₄₇	—	4	4	no	—
90	C₄₇⋊C₈	—	4	4	no	—
91	#20791	—	47	1	no	maximal, Sylow
92	C₂²×C₄₇	C₂²×C₄⋊C₄	1	3	yes	—
93	C₂²×C₄₇	C₂²×M₁₆	1	3	yes	—
94	C₂²×C₄₇	C₂²×M₁₆	1	1	yes	characteristic
95	C₂×D₄₇	—	2	6	no	—
96	C₂×D₄₇	—	2	6	no	—
97	C₂×D₄₇	#384	1	4	yes	—
98	C₄×C₄₇	—	2	6	no	—
99	C₄×C₄₇	#423	1	12	yes	complemented
100	C₄×C₄₇	C₂×#127	1	1	yes	characteristic
101	C₄×C₄₇	C₂×C₄⋊Q₈	1	3	yes	—
102	Dic₄₇	—	2	6	no	—
103	Dic₄₇	#384	1	4	yes	—
104	Dic₄₇	#423	1	12	yes	complemented
105	C₂×#423	—	47	6	no	—
106	C₂×C₄³	—	47	1	no	—
107	#1842	—	47	6	no	—
108	#1842	—	47	1	no	—
109	#1842	D₄₇	1	1	yes	characteristic, complemented
110	C₂×C₄₇	C₂×#384	1	1	yes	characteristic
111	C₂×C₄₇	C₂×#423	1	3	yes	—
112	C₂×C₄₇	C₂×#423	1	3	yes	—
113	D₄₇	—	2	4	no	—
114	C₂²×M₁₆	—	47	4	no	—
115	C₂²×C₄²	—	47	3	no	—
116	C₂²×C₄²	—	47	3	no	—
117	C₂²×C₄²	—	47	1	no	—
118	#423	—	94	24	no	—
119	C₂×C₄⋊C₈	—	47	12	no	—
120	C₂×C₄⋊C₈	—	47	6	no	—
121	C₂×C₄⋊C₈	C₂×D₄₇	1	6	yes	—
122	C₄³	—	47	1	no	—
123	C₄³	—	47	3	no	—
124	C₄³	—	47	3	no	—
125	C₄³	C₂×D₄₇	1	1	yes	characteristic
126	C₄₇	#20791	1	1	yes	characteristic, complemented, Sylow
127	C₂³×C₄	—	47	1	no	—
128	C₂³×C₄	—	47	3	no	—
129	C₂³×C₄	—	47	3	no	—
130	C₂×M₁₆	—	94	24	no	—
131	C₂²×C₈	—	47	4	no	—
132	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
133	C₂×C₄²	—	47	3	no	—
134	C₂×C₄²	—	47	3	no	—
135	C₂×C₄²	—	47	3	no	—
136	C₂×C₄²	—	47	3	no	—
137	C₂×C₄²	—	47	12	no	—
138	C₂×C₄²	—	47	12	no	—
139	C₂×C₄²	—	47	6	no	—
140	C₂×C₄²	—	47	3	no	—
141	C₂×C₄²	—	94	12	no	—
142	C₂×C₄²	—	94	4	no	—
143	C₂×C₄²	C₄×D₄₇	1	3	yes	—
144	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
145	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
146	C₄⋊C₈	—	2	12	no	—
147	C₄⋊C₈	—	94	24	no	—
148	C₄⋊C₈	—	94	12	no	—
149	C₂⁴	—	47	1	no	—
150	C₂²×C₄	—	47	1	no	—
151	C₂²×C₄	—	47	3	no	—
152	C₂²×C₄	—	47	6	no	—
153	C₂²×C₄	—	47	6	no	—
154	C₂²×C₄	—	47	3	no	—
155	C₂²×C₄	—	47	12	no	—
156	C₂²×C₄	—	47	12	no	—
157	C₂²×C₄	—	94	12	no	—
158	C₂²×C₄	—	94	6	no	—
159	C₂²×C₄	—	94	6	no	—
160	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
161	C₂²×C₄	C₂×C₄×D₄₇	1	3	yes	—
162	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
163	M₁₆	—	188	16	no	—
164	C₂×C₈	—	2	12	no	—
165	C₂×C₈	—	94	12	no	—
166	C₄²	—	2	6	no	—
167	C₄²	—	47	12	no	—
168	C₄²	—	47	12	no	—
169	C₄²	—	47	12	no	—
170	C₄²	—	47	6	no	—
171	C₄²	—	47	6	no	—
172	C₄²	—	94	12	no	—
173	C₄²	—	94	6	no	—
174	C₄²	#10625324	1	4	yes	complemented
175	C₄²	D₄×D₄₇	1	6	yes	—
176	C₄²	Q₈×D₄₇	1	6	yes	—
177	C₂³	—	47	6	no	—
178	C₂³	—	94	3	no	—
179	C₂³	—	94	1	no	—
180	C₂³	C₂²×C₄×D₄₇	1	1	yes	characteristic
181	C₂×C₄	—	2	6	no	—
182	C₂×C₄	—	47	6	no	—
183	C₂×C₄	—	47	6	no	—
184	C₂×C₄	—	47	12	no	—
185	C₂×C₄	—	47	4	no	—
186	C₂×C₄	—	47	12	no	—
187	C₂×C₄	—	47	6	no	—
188	C₂×C₄	—	94	24	no	—
189	C₂×C₄	—	94	3	no	—
190	C₂×C₄	—	94	3	no	—
191	C₂×C₄	—	94	24	no	—
192	C₂×C₄	C₄⋊C₄×D₄₇	1	6	yes	—
193	C₂×C₄	C₄⋊C₄×D₄₇	1	6	yes	—
194	C₂×C₄	M₁₆×D₄₇	1	6	yes	—
195	C₂×C₄	C₂×#10625324	1	6	yes	—
196	C₂×C₄	C₂×D₄×D₄₇	1	3	yes	—
197	C₂×C₄	C₂×Q₈×D₄₇	1	3	yes	—
198	C₈	—	4	4	no	—
199	C₈	—	188	4	no	—
200	C₂²	—	47	4	no	—
201	C₂²	—	94	6	no	—
202	C₂²	—	94	6	no	—
203	C₂²	C₂×C₄⋊C₄×D₄₇	1	3	yes	—
204	C₂²	C₂×M₁₆×D₄₇	1	3	yes	—
205	C₂²	C₂²×#10625324	1	1	yes	characteristic
206	C₄	—	2	6	no	—
207	C₄	—	47	4	no	—
208	C₄	—	47	12	no	—
209	C₄	—	94	6	no	—
210	C₄	#15689145	1	12	yes	complemented
211	C₄	#127×D₄₇	1	1	yes	characteristic
212	C₄	C₄⋊Q₈×D₄₇	1	3	yes	—
213	C₂	—	94	4	no	—
214	C₂	#384×D₄₇	1	1	yes	characteristic
215	C₂	#423×D₄₇	1	3	yes	—
216	C₂	C₂×#15689145	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂³×C₄
Fitting	C₄₇×#1842	C₂
prime radical	C₂³×D₄₇	C₂²×C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#20791
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1842

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20243540	C₂	semidirect
4. #20243539	C₂	semidirect
5. C₄₇×#1842	C₂	semidirect
56. C₄²×C₄₇	M₁₆	semidirect
63. C₄×Dic₄₇	M₁₆	semidirect
99. C₄×C₄₇	#423	semidirect
104. Dic₄₇	#423	semidirect
109. #1842	D₄₇	semidirect
126. C₄₇	#20791	semidirect
174. C₄²	#10625324	semidirect
210. C₄	#15689145	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#15689145	2
C₄³×D₄₇	2
#20243540	2
#20243539	2
C₄₇×#1842	2
#20791	47